65 chapter three Heights Coordinates for latitude and longitude, northing and easting, and radius vector and polar angle often come in pairs, but that is not the whole story. For a coordinate pair to be entirely accurate, the point it represents must lie on a well-defined surface. It might be a flat plane, or it might be the surface of a particular ellipsoid; in either case, the surface will be smooth and have a definite and complete mathematical definition. As mentioned earlier, modern geodetic datums rely on the surfaces of geocentric ellipsoids to approximate the surface of the Earth. However, the actual Earth does not coincide with these nice, smooth surfaces, even though that is where the points represented by the coordinate pairs lay. In other words, the abstract points are on the ellipsoid, but the physical features those coordinates intend to represent are, of course, on the actual Earth. Although the intention is for the Earth and the ellipsoid to have the same center, the surfaces of the two figures are certainly not in the same place. There is a distance between them. The distance represented by a coordinate pair on the reference ellipsoid to the point on the surface of the Earth is measured along a line perpendicular to the ellipsoid. This distance is known by more than one name. Known as both the ellipsoidal height and the geodetic height , it is usually symbolized by h . In Figure 3.1 the ellipsoidal height of station Youghall is illustrated. The reference ellipsoid is GRS80 since the latitude and longitude are given in NAD83 (1992). Notice that it has a year in parentheses, 1992. Because 1986 is part of the maintenance of the reference frame for the U.S., the National Spatial Reference System (NSRS), NGS has been updating the calculated horizontal and ellipsoidal height values of NAD83. They differentiate earlier adjustments of NAD83 from those that supersede them by labeling each with the year of the adjustment in parentheses. In other words, NAD83 (1992) supersedes NAD83 (1986). The concept of an ellipsoidal height is straightforward. A reference ellip- soid may be above or below the surface of the Earth at a particular place. If the ellipsoid’s surface is below the surface of the Earth at the point, the ellipsoidal height has a positive sign; if the ellipsoid’s surface is above the TF1625_C03.fm Page 65 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC 66 Basic GIS Coordinates surface of the Earth at the point, the ellipsoidal height has a negative sign. It is important, however, to remember that the measurement of an ellipsoidal height is along a line perpendicular to the ellipsoid, not along a plumb line. Said another way, an ellipsoidal height is not measured in the direction of gravity. It is not measured in the conventional sense of down or up. As explained in Chapter 1, down is a line perpendicular to the ellipsoidal surface at a particular point on the ellipsoidal model of the Earth. On the real Earth down is the direction of gravity at the point. Most often they are not the same, and since a reference ellipsoid is a geometric imagining, it is quite impossible to actually set up an instrument on it. That makes it tough to measure ellipsoidal height using surveying instruments. In other words, ellipsoidal height is not what most people think of as an elevation. Nevertheless, the ellipsoidal height of a point is readily determined using a GPS receiver. GPS can be used to discover the distance from the geocenter of the Earth to any point on the Earth, or above it, for that matter. It has the capability of determining three-dimensional coordinates of a point in a short time. It can provide latitude and longitude, and if the system has the parameters of the reference ellipsoid in its software, it can calculate the ellipsoidal height. The relationship between points can be further expressed in the ECEF coordinates, x, y and z, or in a Local Geodetic Horizon System (LHGS) of north, east, and up. Actually, in a manner of speaking, ellipsoidal heights are new, at least in common usage, since they could not be easily determined until GPS became a practical tool in the 1980s. However, ellip- soidal heights are not all the same, because reference ellipsoids, or sometimes Figure 3.1 Ellipsoidal height. Latitude 40°25'33.39258" Longitude 108 °45'57.78374" Latitude 40°25'33.39258" Longitude 108 °45'57.78374" Ellipsoidal Height +2644.0 Meters a.k.a. Geodetic Height Perpendicular to the Ellipsoidal TF1625_C03.fm Page 66 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC Chapter three: Heights 67 just their origins, can differ. For example, an ellipsoidal height expressed in ITRF97 would be based on an ellipsoid with exactly the same shape as the NAD83 ellipsoid, GRS80; nevertheless, the heights would be different because the origin has a different relationship with the Earth’s surface (see Figure 3.2). Yet there is nothing new about heights themselves, or elevations, as they are often called. Long before ellipsoidal heights were so conveniently avail- able, knowing the elevation of a point was critical to the complete definition of a position. In fact, there are more than 200 different vertical datums in use in the world today. They were, and still are, determined by a method of measurement known as leveling. It is important to note, though, that this process measures a very different sort of height. Both trigonometric leveling and spirit leveling depend on optical instru- ments. Their lines of sight are oriented to gravity, not a reference ellipsoid. Therefore, the heights established by leveling are not ellipsoidal. In fact, a reference ellipsoid actually cuts across the level surfaces to which these instruments are fixed. Two techniques Trigonometric leveling Finding differences in heights with trigonometric leveling requires a level optical instrument that is used to measure angles in the vertical plane, a graduated rod, and either a known horizontal distance or a known slope distance between the two of them. As shown in Figure 3.3, the instrument is centered over a point of known elevation and the rod is held vertically on Figure 3.2 All ellipsoidal heights are not the same. h is NAD83 Ellipsoidal Height 1 h 1 h is ITRF97 Ellipsoidal Height 2 h 2 TF1625_C03.fm Page 67 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC 68 Basic GIS Coordinates the point of unknown elevation. At the instrument, one of two angles is measured: either the vertical angle, from the horizontal plane of the instru- ment, or the zenith angle, from the instrument’s vertical axis. Either angle will do. This measured angle, together with the distance between the instru- ment and the rod, provides two known components of the right triangle in the vertical plane. It is then possible to solve that triangle to reveal the vertical distance between the point at the instrument and the point on which the rod is held. For example, suppose that the height, or elevation, of the point over which the instrument is centered is 100.00 ft. Further suppose that the height of the instrument’s level line of sight, its horizontal plane, is 5.53 ft above that point. Then the height of the instrument (HI) would then be 105.53 ft. For convenience, the vertical angle at the instrument could be measured to 5.53 ft on the rod. If the measured angle is 1º 00' 00" and the horizontal distance from the instrument to the rod is known to be 400.00 ft, all the elements are in place to calculate a new height. In this case, the tangent of 1º 00' 00" multiplied by 400.00 ft yields 6.98 ft. That is the difference in height from the point at the instrument and the point at the rod. Therefore, 100.00 ft plus 6.98 ft indicates a height of 106.98 at the new station where the rod was placed. This process involves many more aspects, such as the curvature of the Earth and refraction of light, that make it much more complex in practice than it is in this illustration. However, the fundamental of the procedure is the solution of a right triangle in a vertical plane using trigonometry, hence the name trigonometric leveling. It is faster and more efficient than spirit leveling but not as precise (more about that later in this chapter). Horizontal surveying usually precedes leveling in control networks. That was true in the early days of what has become the national network of the U.S., the NSRS. Geodetic leveling was begun only after triangulation networks were under way. This was also the case in many other countries. In some places around the world, the horizontal work was completed even before leveling was commenced. In the U.S., trigonometric leveling was applied to geodetic surveying before spirit leveling. Trigonometric leveling was used extensively to provide elevations to reduce the angle observations and base lines necessary to complete triangulation networks to sea level (you will learn more about that in the Sea Level section). The angular measure- ments for the trigonometric leveling were frequently done in an independent operation with instruments having only a vertical circle. Then in 1871 Congress authorized a change for the then Coast Survey under Benjamin Peirce that brought spirit leveling to the forefront. The Coast Survey was to begin a transcontinental arc of triangulation to connect the surveys on the Atlantic Coast with those on the Pacific Coast. Until that time their work had been restricted to the coasts. With the undertaking of trian- gulation that would cross the continent along the 39th parallel, it was clear that trigonometric leveling was not sufficient to support the project. They needed more vertical accuracy than it could provide. So in 1878, at about TF1625_C03.fm Page 68 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC Chapter three: Heights 69 the time the work actually began, the name of the agency was changed from the Coast Survey to U.S. Coast and Geodetic Survey, and a line of spirit leveling of high precision was begun at the Chesapeake Bay heading west. It reached Seattle in 1907. Along the way it provided benchmarks for the use of engi- neers and others who needed accurate elevations (heights) for subsequent work, not to mention establishing the vertical datum for the U.S. Spirit leveling The method shown in Figure 3.4 is simple in principle, but not in practice. An instrument called a level is used to establish a line of sight that is perpendicular to gravity — in other words, a level line. Then two rods marked with exactly the same graduations, like rulers, are held vertically resting on two solid points, one ahead and one behind the level along the route of the survey. The system works best when the level is midway between these rods. When you are looking at the rod to the rear through the telescope Figure 3.3 Trigonometric leveling. Horizontal Elevation of starting point, A. Horizontal distances, d , d between points 12 d 1 d 2 All vertical angles. A C B Elevation of B, C. 400' Measured Height = 5.53' 89° 1° Elev.=100.00' Elev.=106.98' 6.98' H.I. = 5.53' 6 7 5 5 4 5 TF1625_C03.fm Page 69 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC 70 Basic GIS Coordinates of the level, there is a graduation at the point at which the horizontal level line of sight of the level intersects the vertical rod. That reading is taken and noted. This is known as the backsight (BS) . This reading tells the height, or elevation, that the line of sight of the level is above the mark on which the rod is resting. For example, if the point on which the rod is resting is at an elevation of 100 ft and the reading on the rod is 6.78 ft, then the height of the level’s line of sight is 106.78 ft. That value is known as the HI . Then the still level instrument is rotated to observe the vertical rod ahead and a value is read there. This is known as the foresight (FS) . The difference between the two readings reveals the change in elevation from the first point at the BS to the second, at the FS. For example, if the first reading established the height of the level’s line of sight, the HI, at 106.78 ft, and the reading on the rod ahead, the FS, was 5.67 ft, it becomes clear that the second mark is 1.11 ft higher than the first. It has an elevation of 101.11 ft. By beginning this process from a monumented point of known height, a benchmark, and repeating it with good procedures, we can determine the heights of marks all along the route of the survey. Figure 3.4 Spirit leveling. BS =6.78' H.I. =106.78' FS =5.67' Elev.=100.00' Starting Point Elev.=101.11' Elevation of starting point, A. Elevation differences, a b, c, d, etc. Elevation of B, C, and all other points. A C B a b d c TF1625_C03.fm Page 70 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC Chapter three: Heights 71 The accuracy of level work depends on the techniques and the care used. Methods such as balancing the FS and BS, calculating refraction errors, running new circuits twice, and using one-piece rods can improve results markedly. In fact, entire books have been written on the details of proper leveling techniques. Here the goal will be to mention just a few elements pertinent to coordinates generally. It is difficult to overstate the amount of effort devoted to differential spirit level work that has carried vertical control across the U.S. The trans- continental precision leveling surveys done by the Coast and Geodetic Sur- vey from coast to coast were followed by thousands of miles of spirit leveling work of varying precision. When the 39th parallel survey reached the West Coast in 1907, there were approximately 19,700 miles, or 31,789 km, of geodetic leveling in the national network. That was more than doubled 22 years later in 1929 to approximately 46,700 miles, or 75,159 km. As the quantity of leveling information grew, so did the errors and inconsistencies. The foundation of the work was ultimately intended to be Mean Sea Level (MSL) as measured by tide station gauges . Inevitably this growth in leveling information and benchmarks made a new general adjustment of the network necessary to bring the resulting elevations closer to their true values relative to MSL. There had already been four previous general adjustments to the vertical network across the U.S. by 1929. They were done in 1900, 1903, 1907, and 1912. The adjustment in 1900 was based on elevations held to MSL as deter- mined at five tide stations. The adjustments in 1907 and 1912 left the eastern half of the U.S. fixed as adjusted in 1903. In 1927 there was a special adjust- ment of the leveling network. This adjustment was not fixed to MSL at all tide stations, and after it was completed, it became apparent that the MSL surface as defined by tidal observations had a tendency to slope upward to the north along both the Pacific and Atlantic coasts, with the Pacific being higher than the Atlantic. In the adjustment that established the Sea Level Datum of 1929, the determinations of MSL at 26 tide stations (21 in the U.S. and 5 in Canada) were held fixed. Sea level was the intended foundation of these adjustments, and it might make sense to say a few words about the forces that shape it. Evolution of a vertical datum Sea level Both the sun and the moon exert tidal forces on the Earth, but the moon’s force is greater. The sun’s tidal force is about half of that exerted on the Earth by the moon. The moon makes a complete elliptical orbit around the Earth every 27.3 days. There is a gravitational force between the moon and the Earth. Each pulls on the other, and at any particular moment the gravitational pull is greatest on the portion of the Earth that happens to be closest to the moon. That produces a bulge in the waters on the Earth in response to the TF1625_C03.fm Page 71 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC 72 Basic GIS Coordinates tidal force. On the side of the Earth opposite the bulge, centrifugal force exceeds the gravitational force of the Earth and water in this area is forced out away from the surface of the Earth creating another bulge. The two bulges are not stationary, however; they move across the surface of the Earth. They move because not only is the moon moving slowly relative to the Earth as it proceeds along its orbit, but more important, the Earth is rotating in relation to the moon. Because the Earth’s rotation is relatively rapid in comparison with the moon’s movement, a coastal area in the high middle latitudes may find itself with a high tide early in the day when it is close to the moon and with a low tide in the middle of the day when it is has rotated away from it. This cycle will begin again with another high tide a bit more than 24 h after the first high tide. This is because from the moment the moon reaches a particular meridian to the next time it reaches the same meridian, there is actually about 24 h and 50 min, a period called a lunar day. This sort of tide with one high water and one low water in a lunar day is known as a diurnal tide. This characteristic tide would be most likely to occur in the middle latitudes to the high latitudes when the moon is near its maximum declination, as you can see in Figure 3.5. The declination of a celestial body is similar to the latitude of a point on the Earth. It is an angle measured at the center of the Earth from the plane of the equator, positive to the north and negative to the south, to the subject, which is in this case the moon. The moon’s declination varies from its minimum of 0º at the equator to its maximum over a 27.2-day period, and that maximum declination oscillates, too. It goes from ±18.5º up to ±28.5º over the course of an 18.6-year cycle. Figure 3.5 Diurnal tide. (at Maximum North Declination) Declination angle varies from 0 ° to a maximum over 27.2 day period. The maximum declination itself varies +/- 18.5 ° up to +/- 28.5° over the course of an 18.6-year cycle. TF1625_C03.fm Page 72 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC Chapter three: Heights 73 Another factor that contributes to the behavior of tides is the elliptical nature of the moon’s orbit around the Earth. When the moon is closest to the Earth (its perigee) , the gravitational force between the Earth and the moon is 20% greater than usual. At apogee , when the moon is farthest from the Earth, the force is 20% less than usual. The variations in the force have exactly the effect you would expect on the tides, making them higher and lower than usual. It is about 27.5 days from perigee to perigee. To summarize, the moon’s orbital period is 27.3 days. It also takes 27.2 days for the moon to move from its maximum declinations back to 0º directly over the equator. In addition, there are 27.5 days from one perigee to the next. You can see that these cycles are almost the same — almost, but not quite. They are just different enough that it takes from 18 to 19 years for the moon to go through the all the possible combinations of its cycles with respect to the sun and the moon. Therefore, if you want to be certain that you have recorded the full range of tidal variation at a place, you must observe and record the tides at that location for 19 years. This 19-year period, sometimes called the Metonic cycle, is the foun- dation of the definition of MSL. MSL can be defined as the arithmetic mean of hourly heights of the sea at a primary-control tide station observed over a period of 19 years. The mean in Mean Sea Level refers to the average of these observations over time at one place. It is important to note that it does not refer to an average calculation made from mea- surements at several different places. Therefore, when the Sea Level Datum of 1929 was fixed to MSL at 26 tide stations, it was made to fit 26 different and distinct local MSLs. In other words, it was warped to coincide with 26 different elevations. The topography of the sea changes from place to place, which means, for example, that MSL in Florida is not the same as MSL in California. The fact is MSL varies. The water’s temperature, salinity, currents, density, wind, and other physical forces all cause changes in the sea surface’s topography. For example, the Atlantic Ocean north of the Gulf Stream’s strong current is around 1 m lower than it is farther south. The more dense water of the Atlantic is generally about 40 cm lower than the Pacific. At the Panama Canal, the actual difference is about 20 cm from the east end to the west end. A different approach After it was formally established, thousands of miles of leveling were added to the Sea Level Datum of 1929 (SLD29). The Canadian network also contrib- uted data to the SLD29, but Canada did not ultimately use what eventually came to be known as the National Geodetic Vertical Datum of 1929 (NGVD29) . The name was changed on May 10, 1973, because in the end the final result did not really coincide with MSL. It became apparent that the precise leveling done to produce the fundamental data had great internal consistency, but when the network was warped to fit so many tide station determinations of MSL, that consistency suffered. TF1625_C03.fm Page 73 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC 74 Basic GIS Coordinates By the time the name was changed to NGVD29 in 1973, there were more than 400,000 miles of new leveling work included. The network included distortions. Original benchmarks had been disturbed, destroyed, or lost. The NGS thought it time to consider a new adjustment. This time they took a different approach. Instead of fixing the adjustment to tidal stations, the new adjustment would be minimally constrained. That means that it would be fixed to only one station, not 26. That station turned out to be Father Point/ Rimouski, an International Great Lakes Datum of 1985 (IGLD85) station near the mouth of the St. Lawrence River and on its southern bank. In other words, for all practical purposes the new adjustment of the huge network was not intended to be a sea level datum at all. It was a change in thinking that was eminently practical. While is it relatively straightforward to determine MSL in coastal areas, carrying that reference reliably to the middle of a continent is quite another matter. Therefore, the new datum would not be subject to the variations in sea surface topography. It was unimportant whether the new adjustment’s zero elevation and MSL were the same thing or not. The zero point At this stage it is important to mention that throughout the years there were, and continue to be, benchmarks set and vertical control work done by official entities in federal, state, and local governments other than NGS. State depart- ments of transportation, city and county engineering and public works departments, the U.S. Army Corps of Engineers, and many other govern- mental and quasi-governmental organizations have established their own vertical control networks. Included on this list is the U.S. Geological Survey (USGS) . In fact, minimizing the effect on the widely used USGS mapping products was an important consideration in designing the new datum adjust- ment. Several of these agencies, including NOAA, the U.S. Army Corps of Engineers, the Canadian Hydrographic Service, and the Geodetic Survey of Canada, worked together for the development of the IGLD85. This datum was originally established in 1955 to monitor the level of the water in the Great Lakes. Precise leveling proceeded from the zero reference established at Pointe-au-Père, Quebec, Canada, in 1953. The resulting benchmark eleva- tions were originally published in September 1961. The result of this effort was International Great Lakes Datum 1955 . After nearly 30 years, the work was revised. The revision effort began in 1976 and the result was IGLD85 . It was motivated by several developments, including deterioration of the zero reference point gauge location and improved surveying methods. One of the major reasons for the revision, however, was the movement of pre- viously established benchmarks due to isostatic rebound. This effect is liter- ally the Earth’s crust rising slowly, or rebounding , from the removal of the weight and subsurface fluids caused by the retreat of the glaciers from the last ice age. TF1625_C03.fm Page 74 Wednesday, April 28, 2004 10:13 AM © 2004 by CRC Press LLC [...]... (Perpendicular to the Geoid and all Equipotental Surfaces) Geoid (Equipotential Surface) Equipotential Surface Equipotential Surfaces North Converging Earth's Surface Equipotential Surface Ellipsoid Equipotential Surface Mean Sea Level Equipotential ce l Surfa otentia Equip Surface Geoid (Equipotential Surface) Figure 3. 8 Orthometric correction As you can see in Figure 3. 8, the ellipsoid height of a particular... floor In that instance they would all be possessed of the same potential energy from gravity Their potential energies would be equal The floor on which they were resting could be said to be a surface of equal potential, or an equipotential surface © 2004 by CRC Press LLC TF1625_C 03. fm Page 76 Wednesday, April 28, 2004 10: 13 AM 76 Basic GIS Coordinates Now suppose that each of the objects was lifted up... The GEOID 93 model, released at the beginning of 19 93 utilized many more gravity values Both provided geoid heights in a grid of 3 min of latitude by 3 min of longitude, and their accuracy was about 10 cm Next came the GEOID96 model with a grid of 2 min of latitude by 2 min of longitude © 2004 by CRC Press LLC TF1625_C 03. fm Page 83 Wednesday, April 28, 2004 10: 13 AM Chapter three: Heights 83 Today, GEOID99... Howard Rappleye wrote, © 2004 by CRC Press LLC TF1625_C 03. fm Page 79 Wednesday, April 28, 2004 10: 13 AM Chapter three: Heights 79 North Earth's Surface Equipotential Surface Ellipsoid Equipotential Surface Mean Sea Level Equipotential ce l Surfa otentia Equip Surface Geoid (Equipotential Surface) Figure 3. 7 Equipotential surfaces The instruments and methods used in 1878 were continued in use until 1899,... on the same equipotential surface also have the same geopotential numbers along with the same dynamic heights The idea of measuring geopotential by using geopotential numbers was adopted by the IAG in 1955 The geopotential number of a point is the difference between the geopotential below the point, down on the geoid, and the geopotential right at the © 2004 by CRC Press LLC TF1625_C 03. fm Page 84 Wednesday,... Wednesday, April 28, 2004 10: 13 AM 84 Basic GIS Coordinates point itself Said another way, the geopotential number expresses the work that would be done if a weight were lifted from the geoid up to the point, as the weights that were lifted onto a table in the earlier analogy A geopotential number is expressed in geopotential units (gpu) A gpu is 1 kilogal (kgal) per meter Geopotential numbers along with... affected by gravity? a An ellipsoidal height of 2729.4 63 m b 9 83. 124 gals from a gravimeter measurement c The measurement of hourly heights of the sea at a primary-control tide station d An orthometric elevation of 5176.00 ft derived from spirit leveling © 2004 by CRC Press LLC TF1625_C 03. fm Page 86 Wednesday, April 28, 2004 10: 13 AM 86 Basic GIS Coordinates 6 Which of the following events contributed... extent of geoid-ellipsoid separation, also known as the geoidal height, N, at that point © 2004 by CRC Press LLC TF1625_C 03. fm Page 81 Wednesday, April 28, 2004 10: 13 AM Chapter three: Heights 81 Earth's Surface Point on the Earth's Surface Equipotential Surface Equipotential Surface Ellipsoid Height = h (Perpendicular to the Ellipsoid) (A Plumb Line is a Curved Line) Ellipsoid Equipotential Surface... transform the geopotential number that is in kilogals per meter into a dynamic height in meters by dividing by the constant that is in kilogals Here is an example calculation of the dynamic height of station M 39 3, an NGS benchmark: H dyn = H dyn = C g0 1660.419 936 gpu 0.9806199kgals H dyn = 16 93. 235 m The NAVD88 orthometric height of this benchmark determined by spirit leveling is 1694. 931 m and differs...TF1625_C 03. fm Page 75 Wednesday, April 28, 2004 10: 13 AM Chapter three: Heights 75 The choice of the tide gauge at Pointe-au-Père as the zero reference for IGLD was logical in 1955 It was reliable; it had already been connected to the network with precise leveling It was at the outlet of the Great Lakes By 1984, though, the wharf at Pointe-au-Père had deteriorated and the gauge . varies + /- 18.5 ° up to + /- 28.5° over the course of an 18.6-year cycle. TF1625_C 03. fm Page 72 Wednesday, April 28, 2004 10: 13 AM © 2004 by CRC Press LLC Chapter three: Heights 73 Another. suffered. TF1625_C 03. fm Page 73 Wednesday, April 28, 2004 10: 13 AM © 2004 by CRC Press LLC 74 Basic GIS Coordinates By the time the name was changed to NGVD29 in 19 73, there were more than. ellip- soidal heights are not all the same, because reference ellipsoids, or sometimes Figure 3. 1 Ellipsoidal height. Latitude 40°25&apos ;33 .39 258" Longitude 108 °45'57.7 837 4" Latitude